Is there an easier way to cut specifically shaped holes in moderately complex objects (particularly from inside them)?
No Knife!

In STL (the main 3d printing exchange format) everything gets triangulated, converting to quads is a waste of time.

BoolTool has been mentioned, it does add needed functionality to the Boolean Modifer (which can be unintuitive). If you have complicated geometry, curved surfaces with thickness into which you want to cut shapes, then that's a good way to go. But, if like in your image, you have a flat surface then modeling the negative space and snapping it into position and then reconnecting the faces might take the same amount of time.

Take your basic shape:



Then you model the negative space, in place. Learning how to use the snapping tools (vertex snaps in particular) will make modelling easier and speed it up.

I like to remove the object wall, place the negative space where it needs to be, then attach the wall again.





Then select an edge of the remaining opening by holding Alt and Right clicking, this selects the edgeloop if possible (usually it will). Then press F to fill this shape.



With the faces Filled you can press Ctrl+T to triangulate selected faces.



or tidier like yours



At first this may seem like a backwards approach, but when you have simple geometry to cut into the booleans are not the only approach.

Alt + F

docs: Meshes/Editing/Faces

Alt + F is like fill but acknowledges internal boundaries. Islands to exclude from the fill.
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